<< Chapter < Page | Chapter >> Page > |
The half-life of a reaction is the time required to decrease the amount of a given reactant by one-half. The half-life of a zero-order reaction decreases as the initial concentration of the reactant in the reaction decreases. The half-life of a first-order reaction is independent of concentration, and the half-life of a second-order reaction decreases as the concentration increases.
Describe how graphical methods can be used to determine the order of a reaction and its rate constant from a series of data that includes the concentration of A at varying times.
Use the data provided to graphically determine the order and rate constant of the following reaction: ${\text{SO}}_{2}{\text{Cl}}_{2}\phantom{\rule{0.2em}{0ex}}\u27f6\phantom{\rule{0.2em}{0ex}}{\phantom{\rule{0.2em}{0ex}}\text{SO}}_{2}+{\text{Cl}}_{2}$
Time (s) | 0 | 5.00 $\times $ 10 ^{3} | 1.00 $\times $ 10 ^{4} | 1.50 $\times $ 10 ^{4} |
[SO _{2} Cl _{2} ] ( M ) | 0.100 | 0.0896 | 0.0802 | 0.0719 |
Time (s) | 2.50 $\times $ 10 ^{4} | 3.00 $\times $ 10 ^{4} | 4.00 $\times $ 10 ^{4} | |
[SO _{2} Cl _{2} ] ( M ) | 0.0577 | 0.0517 | 0.0415 |
Plotting a graph of ln[SO
_{2} Cl
_{2} ] versus
t reveals a linear trend; therefore we know this is a first-order reaction:
k = −2.20
$\times $ 10
^{5} s
^{−1}
Use the data provided in a graphical method to determine the order and rate constant of the following reaction:
$2P\phantom{\rule{0.2em}{0ex}}\u27f6\phantom{\rule{0.2em}{0ex}}Q+W$
Time (s) | 9.0 | 13.0 | 18.0 | 22.0 | 25.0 |
[P] ( M ) | 1.077 $\times $ 10 ^{−3} | 1.068 $\times $ 10 ^{−3} | 1.055 $\times $ 10 ^{−3} | 1.046 $\times $ 10 ^{−3} | 1.039 $\times $ 10 ^{−3} |
Pure ozone decomposes slowly to oxygen,
${\text{2O}}_{3}(g)\phantom{\rule{0.2em}{0ex}}\u27f6\phantom{\rule{0.2em}{0ex}}{\text{3O}}_{2}(g).$ Use the data provided in a graphical method and determine the order and rate constant of the reaction.
Time (h) | 0 | 2.0 $\times $ 10 ^{3} | 7.6 $\times $ 10 ^{3} | 1.00 $\times $ 10 ^{4} |
[O _{3} ] ( M ) | 1.00 $\times $ 10 ^{−5} | 4.98 $\times $ 10 ^{−6} | 2.07 $\times $ 10 ^{−6} | 1.66 $\times $ 10 ^{−6} |
Time (h) | 1.23 $\times $ 10 ^{4} | 1.43 $\times $ 10 ^{4} | 1.70 $\times $ 10 ^{4} | |
[O _{3} ] ( M ) | 1.39 $\times $ 10 ^{−6} | 1.22 $\times $ 10 ^{−6} | 1.05 $\times $ 10 ^{−6} |
The plot is nicely linear, so the reaction is second order.
k = 50.1 L mol
^{−1} h
^{−1}
From the given data, use a graphical method to determine the order and rate constant of the following reaction:
$2X\phantom{\rule{0.2em}{0ex}}\u27f6\phantom{\rule{0.2em}{0ex}}Y+Z$
Time (s) | 5.0 | 10.0 | 15.0 | 20.0 | 25.0 | 30.0 | 35.0 | 40.0 |
[ X ] ( M ) | 0.0990 | 0.0497 | 0.0332 | 0.0249 | 0.0200 | 0.0166 | 0.0143 | 0.0125 |
What is the half-life for the first-order decay of phosphorus-32? $({}_{15}^{32}\text{P}\phantom{\rule{0.2em}{0ex}}\u27f6\phantom{\rule{0.2em}{0ex}}{}_{16}^{32}\text{S}+{\text{e}}^{-})$ The rate constant for the decay is 4.85 $\times $ 10 ^{−2} day ^{−1} .
14.3 d
What is the half-life for the first-order decay of carbon-14? $({}_{\phantom{\rule{0.5em}{0ex}}6}^{14}\text{C}\u27f6{}_{\phantom{\rule{0.5em}{0ex}}7}^{14}\text{N}+{\text{e}}^{-})$ The rate constant for the decay is 1.21 $\times $ 10 ^{−4} year ^{−1} .
What is the half-life for the decomposition of NOCl when the concentration of NOCl is 0.15 M ? The rate constant for this second-order reaction is 8.0 $\times $ 10 ^{−8} L/mol/s.
8.3 $\times $ 10 ^{7} s
What is the half-life for the decomposition of O _{3} when the concentration of O _{3} is 2.35 $\times $ 10 ^{−6} M ? The rate constant for this second-order reaction is 50.4 L/mol/h.
The reaction of compound A to give compounds C and D was found to be second-order in A . The rate constant for the reaction was determined to be 2.42 L/mol/s. If the initial concentration is 0.500 mol/L, what is the value of t _{1/2} ?
0.826 s
The half-life of a reaction of compound A to give compounds D and E is 8.50 min when the initial concentration of A is 0.150 mol/L. How long will it take for the concentration to drop to 0.0300 mol/L if the reaction is (a) first order with respect to A or (b) second order with respect to A ?
Some bacteria are resistant to the antibiotic penicillin because they produce penicillinase, an enzyme with a molecular weight of 3
$\times $ 10
^{4} g/mol that converts penicillin into inactive molecules. Although the kinetics of enzyme-catalyzed reactions can be complex, at low concentrations this reaction can be described by a rate equation that is first order in the catalyst (penicillinase) and that also involves the concentration of penicillin. From the following data: 1.0 L of a solution containing 0.15 µg (0.15
$\times $ 10
^{−6} g) of penicillinase, determine the order of the reaction with respect to penicillin and the value of the rate constant.
[Penicillin] ( M ) | Rate (mol/L/min) |
---|---|
2.0 $\times $ 10 ^{−6} | 1.0 $\times $ 10 ^{−10} |
3.0 $\times $ 10 ^{−6} | 1.5 $\times $ 10 ^{−10} |
4.0 $\times $ 10 ^{−6} | 2.0 $\times $ 10 ^{−10} |
The reaction is first order.
k = 1.0
$\times $ 10
^{7} mol
^{−1} min
^{−1}
Both technetium-99 and thallium-201 are used to image heart muscle in patients with suspected heart problems. The half-lives are 6 h and 73 h, respectively. What percent of the radioactivity would remain for each of the isotopes after 2 days (48 h)?
There are two molecules with the formula C
_{3} H
_{6} . Propene,
${\text{CH}}_{3}\text{CH}={\text{CH}}_{2},$ is the monomer of the polymer polypropylene, which is used for indoor-outdoor carpets. Cyclopropane is used as an anesthetic:
When heated to 499 °C, cyclopropane rearranges (isomerizes) and forms propene with a rate constant of
5.95
$\times $ 10
^{−4} s
^{−1} . What is the half-life of this reaction? What fraction of the cyclopropane remains after 0.75 h at 499.5 °C?
4.98; 20% remains
Fluorine-18 is a radioactive isotope that decays by positron emission to form oxygen-18 with a half-life of 109.7 min. (A positron is a particle with the mass of an electron and a single unit of positive charge; the equation is ${}_{518}^{\phantom{\rule{1em}{0ex}}9}\text{F}\u27f6{}_{18}^{\phantom{\rule{0.5em}{0ex}}8}\text{O}+{\text{e}}^{-}.)$ Physicians use ^{18} F to study the brain by injecting a quantity of fluoro-substituted glucose into the blood of a patient. The glucose accumulates in the regions where the brain is active and needs nourishment.
(a) What is the rate constant for the decomposition of fluorine-18?
(b) If a sample of glucose containing radioactive fluorine-18 is injected into the blood, what percent of the radioactivity will remain after 5.59 h?
(c) How long does it take for 99.99% of the ^{18} F to decay?
Suppose that the half-life of steroids taken by an athlete is 42 days. Assuming that the steroids biodegrade by a first-order process, how long would it take for $\frac{1}{64}$ of the initial dose to remain in the athlete’s body?
252 days
Recently, the skeleton of King Richard III was found under a parking lot in England. If tissue samples from the skeleton contain about 93.79% of the carbon-14 expected in living tissue, what year did King Richard III die? The half-life for carbon-14 is 5730 years.
Nitroglycerine is an extremely sensitive explosive. In a series of carefully controlled experiments, samples of the explosive were heated to 160 °C and their first-order decomposition studied. Determine the average rate constants for each experiment using the following data:
Initial [C _{3} H _{5} N _{3} O _{9} ] ( M ) | 4.88 | 3.52 | 2.29 | 1.81 | 5.33 | 4.05 | 2.95 | 1.72 |
t (s) | 300 | 300 | 300 | 300 | 180 | 180 | 180 | 180 |
% Decomposed | 52.0 | 52.9 | 53.2 | 53.9 | 34.6 | 35.9 | 36.0 | 35.4 |
[ A ] _{0} ( M ) | k $\times $ 10 ^{3} (s ^{−1} ) |
---|---|
4.88 | 2.45 |
3.52 | 2.51 |
2.29 | 2.54 |
1.81 | 2.58 |
5.33 | 2.35 |
4.05 | 2.44 |
2.95 | 2.47 |
1.72 | 2.43 |
For the past 10 years, the unsaturated hydrocarbon 1,3-butadiene
$({\text{CH}}_{\text{2}}=\text{CH}\u2013\text{CH}={\text{CH}}_{2})$ has ranked 38th among the top 50 industrial chemicals. It is used primarily for the manufacture of synthetic rubber. An isomer exists also as cyclobutene:
The isomerization of cyclobutene to butadiene is first-order and the rate constant has been measured as 2.0 $\times $ 10 ^{−4} s ^{−1} at 150 °C in a 0.53-L flask. Determine the partial pressure of cyclobutene and its concentration after 30.0 minutes if an isomerization reaction is carried out at 150 °C with an initial pressure of 55 torr.
Notification Switch
Would you like to follow the 'Chemistry' conversation and receive update notifications?